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Implicit methods for hyperbolic equations are analyzed using LU decompositions. It is shown that the inversion of the resulting tridiagonal matrices is usually stable even when diagonal dominance is lost. Furthermore, these decompositions can be used to construct stable algorithms in multidimensions. When marching to a steady state, the solution is independent of the time. Alternating direction methods which solve for u n+1 − u n are unconditionally unstable in three-space dimensions and so the new method is more appropriate. Furthermore, only two factors are required even in threespace dimensions and the operation count per time step is low. Acceleration to a steady state is analyzed, and it is shown that the fully implicit method with large time steps approximates a Newton-Raphson iteration procedure.
Jameson et al. (Thu,) studied this question.